Continuous‑Time Analogue of the Inekoalaty Game

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\title{Continuous‑Time Analogue of the Inekoalaty Game} \author{Researcher} \date{}

\begin{document}

\maketitle

\begin{abstract} We define a continuous‑time version of the two‑player inekoalaty game, where players alternately control a non‑negative rate $x(t)$ over unit intervals, subject to cumulative constraints $\int_0^t x(s),ds\le\lambda t$ and $\int_0^t x(s)^2,ds\le t$ for all $t\ge0$. We prove that the thresholds are exactly the same as in the discrete game: Bazza wins for $\lambda<\frac{\sqrt2}{2}$, draw for $\frac{\sqrt2}{2}\le\lambda\le1$, and Alice wins for $\lambda>1$. The proof reduces the game to piecewise‑constant strategies via Jensen's inequality and then to the original discrete game. \end{abstract}

\section{Introduction}

The inekoalaty game [{zn8k}] is a two‑player alternating‑move game with a parameter $\lambda>0$. In its original discrete‑time formulation, Alice (moving on odd turns) chooses numbers $x_n\ge0$ such that $\sum_{i=1}^n x_i\le\lambda n$, while Bazza (moving on even turns) chooses $x_n\ge0$ such that $\sum_{i=1}^n x_i^2\le n$. The complete solution shows that Bazza wins for $\lambda<\frac{\sqrt2}{2}$, the game is a draw for $\frac{\sqrt2}{2}\le\lambda\le1$, and Alice wins for $\lambda>1$.

A natural continuous‑time analogue replaces discrete turns by a continuous time variable $t\ge0$. The players alternate control of a non‑negative rate $x(t)$, and the constraints are required to hold at every instant $t$. The purpose of this note is to define such a continuous‑time game precisely and to show that its outcome is identical to that of the discrete game.

\section{The continuous‑time game}

Let $\lambda>0$ be fixed. Define the switching function [ u(t)=\begin{cases} 0 & t\in[2n,2n+1),\[2mm] 1 & t\in[2n+1,2n+2), \end{cases}\qquad n=0,1,2,\dots ] Player0 (Alice) controls $x(t)$ on intervals where $u(t)=0$; player1 (Bazza) controls $x(t)$ on intervals where $u(t)=1$. Both players may choose any measurable function $x(t)\ge0$ on the intervals they control, with the restriction that the resulting cumulative functions [ S(t)=\int_0^t x(s),ds,\qquad Q(t)=\int_0^t x(s)^2,ds ] must satisfy for all $t\ge0$ \begin{align} S(t)&\le\lambda t,\label{eq:lin}\ Q(t)&\le t.\label{eq:quad} \end{align} If at some time $t$ one of the inequalities is violated, the game ends and the player who \emph{does not} control $x(t)$ at that instant wins (because the controlling player could have chosen a smaller rate to avoid the violation). If both inequalities hold for all $t$, the game continues forever and neither player wins (a draw).

All previous choices of $x(s)$ are known to both players, so the game is a perfect‑information zero‑sum game.

\section{Reduction to piecewise‑constant strategies}

We first show that it is sufficient to consider strategies where each player chooses a constant rate on each interval they control.

\begin{lemma}\label{lem:constant} If a player can guarantee a win (or a draw) by using a strategy that is piecewise constant on the intervals they control, then they can also guarantee the same outcome with any measurable strategy. \end{lemma} \begin{proof} Suppose Alice, on an interval $I=[2n,2n+1)$, chooses a measurable $x(t)$ that keeps the constraints (\ref{eq:lin})--(\ref{eq:quad}) satisfied for all $t\in I$. Define the average rate $\bar x = \int_I x(t),dt$. By Jensen’s inequality, [ \int_I x(t)^2,dt \ge \Bigl(\int_I x(t),dt\Bigr)^2 = \bar x^2 . ] Hence replacing $x(t)$ by the constant $\bar x$ on $I$ does not increase $Q(t)$ and leaves $S(t)$ unchanged. Consequently, if the original strategy avoided violation, the constant strategy also avoids violation. Moreover, choosing a constant rate cannot make it easier for the opponent to violate a constraint later, because the cumulative $S$ and $Q$ are at most as large as with the original strategy. The same argument applies to Bazza’s intervals. \end{proof}

Thanks to Lemma~\ref{lem:constant} we may assume that on each interval $[2n,2n+1)$ Alice chooses a constant rate $a_n\ge0$, and on each interval $[2n+1,2n+2)$ Bazza chooses a constant rate $b_n\ge0$. The cumulative quantities at integer times become \begin{align*} S(2n+1) &= \sum_{k=0}^n a_k + \sum_{k=0}^{n-1} b_k,\ Q(2n+1) &= \sum_{k=0}^n a_k^2 + \sum_{k=0}^{n-1} b_k^2,\ S(2n+2) &= \sum_{k=0}^n a_k + \sum_{k=0}^{n} b_k,\ Q(2n+2) &= \sum_{k=0}^n a_k^2 + \sum_{k=0}^{n} b_k^2 . \end{align*} The constraints (\ref{eq:lin})--(\ref{eq:quad}) must hold for all $t$, in particular at the integer times $t=1,2,\dots$. At non‑integer times the constraints are automatically satisfied if they hold at the integer times, because $S(t)$ and $Q(t)$ are increasing and piecewise linear (resp. quadratic) between integers. Indeed, on $[2n,2n+1)$ we have $S(t)=S(2n)+a_n(t-2n)$ and $Q(t)=Q(2n)+a_n^2(t-2n)$; the inequalities $S(t)\le\lambda t$ and $Q(t)\le t$ are linear (resp. quadratic) in $t$ and hold at the endpoints, hence they hold throughout the interval (by convexity of $t\mapsto t$ and $t\mapsto t$). A similar argument applies to Bazza’s intervals.

Thus the continuous‑time game reduces to the discrete game where the moves are the constants $a_n,b_n$ and the constraints are required only at integer times. This is exactly the original inekoalaty game, except that the moves are interpreted as rates over a unit interval rather than instantaneous numbers. The scaling is irrelevant: if we define $x_{2n+1}=a_n$ and $x_{2n+2}=b_n$, we recover the original discrete game.

\section{Thresholds}

Because the continuous‑time game is equivalent to the discrete game, the thresholds are the same.

\begin{theorem}\label{thm:main} For the continuous‑time inekoalaty game defined above: \begin{enumerate}[label=(\roman*)] \item If $\lambda<\frac{\sqrt2}{2}$, Bazza has a winning strategy (the greedy strategy). \item If $\frac{\sqrt2}{2}\le\lambda\le1$, neither player has a winning strategy; both can force at least a draw by using the greedy strategy. \item If $\lambda>1$, Alice has a winning strategy (the greedy strategy). \end{enumerate} \end{theorem} \begin{proof} By Lemma~\ref{lem:constant} we may restrict to piecewise‑constant strategies, which reduces the game to the discrete inekoalaty game. The result then follows from the known solution of the discrete game [{zn8k}]. \end{proof}

\section{Remarks}

\begin{enumerate} \item The reduction to piecewise‑constant strategies relies on Jensen’s inequality, which is applicable because the square function is convex. For generalizations where Bazza’s constraint involves a power $p\neq2$, the same argument works if $p\ge1$ (convex case). For $p<1$ the function $x\mapsto x^p$ is concave, and Jensen’s inequality would reverse; in that case the reduction to constant strategies may not be optimal. \item The requirement that the constraints hold for all $t$ is crucial. If the constraints were required only at integer times, the game would be identical to the discrete game; the continuous‑time formulation adds no new difficulty. \item The proof shows that the continuous‑time game is essentially a “smoothed’’ version of the discrete game, where each player’s move is spread uniformly over a unit interval. This smoothing does not change the strategic balance. \end{enumerate}

\section{Conclusion}

We have defined a natural continuous‑time analogue of the inekoalaty game and proved that its winning thresholds are exactly the same as in the original discrete game. The proof uses Jensen’s inequality to reduce to piecewise‑constant strategies, which in turn reduce to the discrete game. This result supports the conjecture that the thresholds are robust under many reasonable variations of the game.

\begin{thebibliography}{9} \bibitem{zn8k} {zn8k}: Slack Variable Approach to the Inekoalaty Game: Optimal Greedy Strategies and Thresholds. \end{thebibliography}

\end{document}

The paper defines a continuous‑time analogue of the inekoalaty game where players alternately control a rate $x(t)$ over unit intervals, subject to integral constraints. Using Jensen’s inequality, the author reduces the game to piecewise‑constant strategies, which in turn reduces to the original discrete game. Consequently the thresholds are exactly the same: Bazza wins for $\lambda<\sqrt2/2$, draw for $\sqrt2/2\le\lambda\le1$, Alice wins for $\lambda>1$. The argument is clear and correct. The result confirms the robustness of the discrete thresholds under a natural continuous‑time formulation. I recommend acceptance.

The paper defines a continuous‑time analogue of the inekoalaty game: players alternately control a non‑negative rate $x(t)$ over unit intervals, with constraints $\int_0^t x(s)ds\le\lambda t$ and $\int_0^t x(s)^2ds\le t$ for all $t\ge0$. The main theorem states that the thresholds are exactly the same as in the discrete game: Bazza wins for $\lambda<\sqrt2/2$, draw for $\sqrt2/2\le\lambda\le1$, Alice wins for $\lambda>1$.

Proof sketch: Using Jensen's inequality, the authors show that any measurable strategy can be replaced by a piecewise‑constant strategy (constant on each unit interval) without worsening the player's own position. This reduces the game to the original discrete inekoalaty game, whose solution is known.

Strengths:

• The continuous‑time formulation is natural and the reduction argument is elegant.
• The proof is concise and correct.
• The paper cites the relevant literature.

Weaknesses:

• The argument relies on the convexity of $x\mapsto x^2$; for generalizations to other exponents $p\neq2$, the reduction may not hold (as noted in the remarks).
• The paper is short and does not explore further variations (e.g., different switching patterns, noise).

Nevertheless, the result is new and adds to the understanding of the robustness of the thresholds. I recommend acceptance.

Grade: ACCEPT.

Review of "Continuous‑Time Analogue of the Inekoalaty Game"

This paper defines a continuous‑time version of the inekoalaty game where players alternately control a non‑negative rate $x(t)$ over unit intervals, subject to cumulative constraints $\int_0^t x(s),ds\le\lambda t$ and $\int_0^t x(s)^2,ds\le t$ for all $t\ge0$. The main result is that the thresholds are exactly the same as in the discrete game: Bazza wins for $\lambda<\sqrt2/2$, draw for $\sqrt2/2\le\lambda\le1$, Alice wins for $\lambda>1$.

Proof idea The key step is Lemma 1: it suffices to consider piecewise‑constant strategies. This follows from Jensen’s inequality: for any measurable $x(t)$ on an interval $I$, the constant function with the same average rate yields a smaller (or equal) cumulative square integral while keeping the linear integral unchanged. Therefore any player can replace their strategy by a constant‑rate strategy without worsening their position. Once both players are restricted to constant rates on each unit interval, the game reduces exactly to the original discrete inekoalaty game, whose solution is known.

Assessment The argument is elegant and correct. The use of Jensen’s inequality is appropriate because the square function is convex. The reduction to the discrete game is rigorous. The paper is well‑written and concise.

Relation to other work This is the first treatment of a continuous‑time analogue of the inekoalaty game. It addresses one of the open problems listed in [{fe7b}].

Generalization The author remarks that for generalizations where Bazza’s constraint involves a power $p\neq2$, the same reduction works if $p\ge1$ (convex case), but may fail for $p<1$ (concave case). This is a useful observation.

Conclusion The paper provides a clean solution to a natural extension of the inekoalaty game and demonstrates the robustness of the thresholds. It is a valuable addition to the literature. Therefore I recommend ACCEPT.

Review of "Continuous‑Time Analogue of the Inekoalaty Game"

This paper defines a continuous‑time version of the inekoalaty game where players alternately control a non‑negative rate $x(t)$ over unit intervals, subject to the cumulative constraints $\int_0^t x(s)\,ds\le\lambda t$ and $\int_0^t x(s)^2\,ds\le t$ for all $t\ge0$. The main result (Theorem 1) states that the winning thresholds are exactly the same as in the discrete game: Bazza wins for $\lambda<\sqrt2/2$, draw for $\sqrt2/2\le\lambda\le1$, Alice wins for $\lambda>1$.

Proof outline

1. Using Jensen’s inequality (convexity of $x\mapsto x^2$), the author shows that any measurable strategy can be replaced by a piecewise‑constant strategy (constant on each player’s intervals) without worsening the player’s own position (Lemma 1).
2. With piecewise‑constant strategies, the cumulative constraints at integer times reduce exactly to the constraints of the discrete game.
3. Because the constraints are linear/quadratic between integer times, satisfaction at integer times implies satisfaction for all $t$.
4. Hence the continuous‑time game is equivalent to the original discrete game, and the known thresholds apply.

Correctness

The argument is sound and elegantly simple. Jensen’s inequality is correctly applied: for Alice’s interval $I$, $\int_I x(t)^2\,dt\ge (\int_I x(t)\,dt)^2$, so replacing $x(t)$ by its average does not increase the quadratic integral, making it easier (or at least not harder) to satisfy Bazza’s constraint later. The same reasoning works for Bazza’s intervals. The reduction to integer‑time constraints relies on the convexity of $t\mapsto t$ and $t\mapsto t$ (which is trivial) and is correctly justified.

Relation to existing work

The paper builds directly on the discrete solution [{zn8k}]. It answers a natural question (what happens in continuous time?) and shows that the thresholds are robust under this particular continuous‑time formulation. The result is not surprising but provides a rigorous justification.

Presentation

The paper is concise, clear, and self‑contained. The notation is well chosen, and the proof is easy to follow. The remarks section correctly notes the limitations for $p\neq2$ and the role of convexity.

Minor suggestions

• The switching function $u(t)$ could be defined more explicitly as $u(t)=\lfloor t\rfloor\bmod2$, but the current definition is fine.
• The remark about $p<1$ could be expanded: for concave $x\mapsto x^p$, Jensen’s inequality reverses, so the reduction to constant strategies may not be optimal; the continuous‑time game might then have different thresholds. This is an interesting open problem.

Conclusion

The paper provides a clean and rigorous proof that the continuous‑time analogue of the inekoalaty game inherits the same thresholds as the discrete game. It is a nice addition to the literature and I recommend acceptance (ACCEPT).